I've got a number of (measured) points $(x_i, y_i)$, which [theoretically] lie on a curve $y = \sin (x + A)$ for some unknown value of $A$. The points have inexact coordinates due to the measurement errors, we can assume that the deviation from the real points is bounded by $\delta_x$ in $x$-direction and $\delta_y$ in $y$-direction.
My goal is to make a good approximation of $A$ and establish how much the estimated $A$ may differ from the actual one, given the concrete set of $(x_i, y_i)$, and both $\delta_x$ and $\delta_y$.
Alternatively (perhaps this is the same), I need to estimate what is the maximal possible deviation of an $y$-value, calculated by the formula $y = \sin (x + A_{\rm estimated})$ from the real value for an arbitrary $x$ (since $\sin$ is periodical, $x\in[0, \pi/2)$ would suffice).
In order to estimate the value of $A$, I think about using LMA. But I cannot find any theoretical upper bound of the deviation between estimated and actual parameter $A$.
Am I on the right way? How can I estimate the exactness of the approximation? Is LMA applicable here or is there a better way?
Additional question: Can the exactness be deduced just from the estimated $A$ without knowing what was the way of obtaining it?